They obsess over fat content because there is so much misinformation about weight loss. Heck, even this thread is full of misinformation.
There was a simple gimmick a few decades back on an infomercial. That catch phrase was "fat makes you fat". Many years have passed, the product is long gone, yet we can't ever end that marketing blurb.
The result: instead of eating a handfull of full fat chips (~3 oz of weight, 30g fat, 450 calories), they now choose a full bag of reduced fat chips (6 oz of weight, 35g fat, 800 calories). The more low-fat foods they consume, the more they convince themselves that they can overeat, and the more weight they gain. Low-fat diets almost always end up being high-mass and high-calorie diets.
On the surface, there isn't anything wrong with a lower fat diet (assuming you don't go too low). The problem is that people can't ever truely stick to a lower fat diet. They think they are sticking to it, but they aren't.